Talk:Inner product space

Ordered Field

Doesnt the field the inner product maps to have to be an ordered field? What kind of order? Doesnt make sense to talk about inequalities (positive definiteness) in a non-ordered field, so I dont know why complex numbers are an appropriate field.50.35.97.238 (talk) 20:12, 6 November 2018 (UTC)

Update: Im assuming that positive definiteness between an element and itself <x,x> must map to an ordered subfield of the complex then, namely (most likely) the reals? This was not specified. 50.35.97.238 (talk) 20:17, 6 November 2018 (UTC)

Negative or zero

This definition is too narrow. Inner products may be negative or zero. The definition given is for a positive definite inner product.

That is a very context-dependent point. In many contexts, inner product is indeed taken to mean a positive-definite bilinear form. Michael Hardy 02:17, 8 Oct 2004 (UTC)

Yes, bilinear form is the term to use instead of "indefinite inner product".My primary concern was Minkowski space which is handled with the bilinear form concept even if there is a tradition of persisting in the inner product terminology.Rgdboer 23:26, 13 September 2006 (UTC)

To illustrate that, and how, the Triangle inequality is obtained as a consequence of the Cauchy-Bunyakovski-Schwarz inequality:

Priliminaries, directly from the axioms:

<1, 1> > 0.

<(<x, y> - <y, x>), (<x, y> - <y, x>)> >= 0,

<(<x, y> - <y, x>), (<x, y> - <y, x>)> =

(<y, x> - <x, y>) <1, (<x, y> - <y, x>)> =

-(<x, y> - <y, x>) <1, (<x, y> - <y, x>)> =

-(<x, y> - <y, x>) <1, 1> (<x, y> - <y, x>) =

-(<x, y> - <y, x>)^2 <1, 1> >= 0; therefore

(<x, y> - <y, x>)^2 =< 0.

Now adding the Cauchy-Bunyakovski-Schwarz inequality:

4 <x, y> <y, x> =< 4 <x, x> <y, y>:

(<x, y> - <y, x>)^2 + 4 <x, y> <y, x> =< 4 <x, x> <y, y>,

(<x, y> + <y, x>)^2 =< 4 <x, x> <y, y>.

According to preliminaries also: 0 =< (<x, y> + <y, x>)^2. Together with axioms, therefore

(<x, y> + <y, x>) =< 2 Sqrt( <x, x> <y, y> ),

(<x, y> + <y, x>) + <x, x> + <y, y> =< 2 Sqrt( <x, x> <y, y> ) + <x, x> + <y, y>,

<(x + y), (x + y)> =< (Sqrt( <x, x> ) + Sqrt( <y, y> ))^2

Finally, per axioms 0 =< <(x + y), (x + y)>; thereby

Sqrt( <(x + y), (x + y)> ) =< Sqrt( <x, x> ) + Sqrt( <y, y> ),

i.e. || x + y || =< || x || + || y ||.

Regards, Frank W ~@) R 02:13 Mar 3, 2003 (UTC).

In addition to this the definition of non-degeneracy is misleading: a non-degenerate inner product is one for which the matrix that generates it is non-singular. An hermitian matrix of signature (n,1) has an entire subspace of zero vectors in C^{n+1}, but the hermitian form is non-degenerate. The correct definition is http://planetmath.org/encyclopedia/NonDegenerate.html —Preceding unsigned comment added by 82.13.19.79 (talk) 14:54, 17 December 2007 (UTC)

Sequilinearity

Though I am used to Sequilinearity as measing linear in 1st argument, most physicists use the other convention. I have edited some articles with this assumption; moreover, many of the quantum mechanics articles naturally follow the physicist's convention. At this point the physicist's convention seems the one to follow in Wikipedia.CSTAR 14:07, 11 Aug 2004 (UTC)

Agreed. There's some confusion, but I think the physicist's convention is the one to follow, since it has a good reason for using that order. The convention seems to be working its way into recent mathematics, as well. -FunnyMan 18:28, Oct 23, 2004 (UTC)

I, and I belive most mathematicians, regard the inner product as being linear in the 1st argument. While I appreciate that this may be inconsistent with the way physicists use the inner product, the article seems to be written as if it were from a mathematical viewpoint, and so I think it should either be defined in the mathematical way, or it should be made clear that this is a definition for physics and that mathematicians use another convention. As to the suggestion that this "convention seems to be working its way into recent mathematics", even if this is so, it is certainly not standard practice yet (for example, most university mathematics courses, and every textbook I have seen, would define the innner product as being linear in the 1st argument), and if I understand Wikipedia's ethos correctly, the article should reflect current practice.

Frankly I dont see a point to the discussion in the first place. I dont know why this is a notable topic. Given the conjugate symmetry property and linearity in either term, you can easily prove linearity in the other term. Its arbitrary and moot. I get the sense that you guys are approaching this issue from the perspective of a physicist who only knows implementation, rather than the perspective of a mathematician who abstracts. Im less concerned with what is more popular and would prefer to default definition to the historical inspiration. 50.35.97.238 (talk) 17:52, 6 November 2018 (UTC)

Bilinear?

From the definitions of inner product and bilinear operator, follows that the inner product over a complex vector space isn't a bilinear form (but it is for real vector spaces). So, the definition in the article contains a wrong generalization:

"Formally, an inner product space is a vector space V over the field F together with a bilinear form, called an inner product"

Good point. We could split it into two cases, or we could rewrite sesquilinear form to also apply to R. -- Walt Pohl 07:25, 12 Jan 2005 (UTC)

Fixed. -- Walt Pohl 01:00, 13 Jan 2005 (UTC)

I think the inner product map should be V × V → F and not V × V → R. At least the standard inner product on C is not real-valued. I changed the article accordingly. -- Jitse Niesen 12:00, 13 Jan 2005 (UTC)

Oops, you're right. -- Walt Pohl 03:37, 14 Jan 2005 (UTC)

I have never seen a definition of inner product which isn't linear in the first argument. I am a mathematician and consider the current version incorrect.

Connnection to Bra-Ket Notation

Could we add an explanation on how the inner-product is related to Dirac notation? Perhaps I am requesting a discussion on the notion of dual spaces and inner-products. In "Principles of Quantum Mechanics," Shankar mentions that one can think of the inner-product as a mapping from V* × V rather than a mapping from V × V where V* is the dual space of bras (obtained by taking the conjugate transpose of the kets).

Also, it seems that we should mention another common notation for inner-products: (x,y). In fact, the page on bra-ket notation refers to the inner-product in this manner. (User:--- forgot to sign)

You can surely add one more section to the article explaining the connection to bra-ket notation. Oleg Alexandrov 15:30, 5 Jun 2005 (UTC)

Separate inner product page?

It seems to me that the current structure of the inner product space and dot product articles could be improved. Inner product redirects to dot product, which seems inaccurate since the dot product is just an example of the inner product but is not synonymous. Also, the majority of the inner product space article describes the properties of the inner product. Wouldn't it make sense to move that content to a separate inner product article? Vanished user 1029384756 20:10, 20 July 2005 (UTC)

Let me start by saying that it is nice to see some fresh blood. Your contributions to the linear algebra articles are much appreciated. Are you already aware of Wikipedia:WikiProject Mathematics?

To return to your question: it seems from the top of both articles that the terms inner product and dot product are treated as synonymous. I actually quite like the current division: dot product describes the standard inner product on Euclidean space and inner product space describes for general inner products. The articles can be much improved, though, especially dot product.

Why do you want to split this article? I think it would be rather difficult to do it cleanly, and the article is not that big that it needs to be split. -- Jitse Niesen (talk) 20:41, 20 July 2005 (UTC)

I agree with Jitse Niesen.--CSTAR 21:17, 20 July 2005 (UTC)

Wow, speedy resposes. Thanks for directing me to Wikipedia:WikiProject Mathematics, I wasn't aware of it.

I guess my problem is that inner product and dot product are treated as synonomous. In Euclidean space they often are treated as synonymous but in the general definition, the dot product is not synonymous and is just an example of an inner product. At least that's what I learned. Maybe splitting the article isn't the best solution, but I feel that some clarification is needed at least in the beginning paragraphs. Perhaps inner product should redirect to inner product space instead? Perhaps inner product should not be a redirect at all but have some content that explains the axioms and properties? Maybe inner product space should move to inner product? Personally I like the way MathWorld has these topics organized with an Inner Product page, a separate Dot Product page, and a minimal Inner Product Space page. Just some thoughts... -Vanished user 1029384756 22:30, 20 July 2005 (UTC)

I think it is correct to say that dot product refers only to a specific case (but the informal verb "to dot with" can also be used with general inner products), so please do edit the articles to make this clear. I don't like short pages like the MathWorld inner product space page, unless they have potential for growth, and this seems to be the general opinion here on Wikipedia. -- Jitse Niesen (talk) 18:35, 22 July 2005 (UTC)

Vote for new external link

Here is my site with inner product example problems. Someone please put this link in the external links section if you think it's helpful and relevant. Tbsmith

http://www.exampleproblems.com/wiki/index.php/Linear_Algebra#Inner_Products

an inner product (also called scalar product or dot product),

In the first paragraph we have an inner product (also called scalar product or dot product), .... Are all inner products dot products, or is the dot product an example of an inner product? --Salix alba (talk) 16:45, 19 February 2006 (UTC)

I thought those things were synonymous. I did not hear of the notion of "inner product" in other settings. 23:06, 19 February 2006 (UTC)

I was under the impression that the dot product was a specific example of an inner product -- that seems most consistent with dot product, and the definition of dot product given in the Examples section. Also, I know computer scientists sometimes talk about dot product, referring to the "specific example" definition (i.e, $\sum _{i=0}^{n}x_{n}{\overline {y_{n}}}$), as it's often important for a system to be able to perform this particular operation quickly -- system performance is sometimes measured in "dot products per second". I'd be in favour of rewording the intro to say that these things are related, and not simply the same thing. James pic (talk) 11:00, 18 June 2008 (UTC)

What map?

The canonical map isn't defined anywhere.

The map from V to the dual space V* is an isomorphism. For a finite-dimensional vector space, it suffices to check injectivity:

Moreover, it doesn't say that the map is in some way associated to the sesquilinear form.--CSTAR 18:05, 9 March 2006 (UTC)

I guess I'm the one responsible for that glaring ommission. All I can say in my defense is that I had a complete rewrite of this article in a browser window which crashed, completely draining my impetus. I guess I have to fix it. -lethe ^{talk +} 04:24, 24 April 2006 (UTC)

I think the article is in a bad state, and it needs to be restructured, but for now, I've given the map. -lethe ^{talk +} 05:49, 24 April 2006 (UTC)

First a minor detail, the map should be $x\mapsto \langle \cdot ,x\rangle$ if the product is linear in the first argument, and it is an antilinear isomorphism, not an isomorphism.

Anyway, I think requiring it to be an iso is too strong because the dual space is always complete, so V would already have to be a Hilbert space. And in a Hilbert space, the surjectivity of the map is actually the content of the Riesz representation theorem, not part of the definition.

I will change the article, correct me if I'm wrong.

I'm not criticising you lethe, I'd probably be annoyed too after a browser crash :)

Functor salad 19:17, 21 July 2007 (UTC)

Vector Notation (?)

I know that a lot of mathematicians do not use an arrow to denote a non-spacial vector, but in just about every textbook I've read, vectors (even non-spatial ones) are always boldfaced. Is this something we should change, or does nobody care?--Sick0Fant 01:05, 24 July 2006 (UTC)Φ

I suppose nobody cares... if you work for some time in vector spaces (which often will be spaces of functions defined on other vector spaces), you sooner or later stop putting arrows on all these vectors - actually it would rather confuse me to see arrows on functions, but spaces of functions are of course vector spaces (usually), so functions are vectors (and vectors are functions, in fact). I think that when you start using the vocabulary of "inner product space", you are usually in a case where almost everything is a vector and only very few objects are scalars, which you then denote by Greek letters since too much boldface or arrows would be more confusing than helpful. (Boldface is then sometimes rather used for matrices of scalars and the special case of coordinate "vectors", i.e. column matrices.) (In some rare cases, a little lack of notational consistency can be more pedagogical than too much notational rigidity.) However, this article is somewhere in the middle between these two worlds, and if you wish to put all vectors in boldface, please don't hesitate and do it! — MFH:Talk 23:03, 4 September 2007 (UTC)

They really should put the vectors in boldface to distinguish them from scalars. I know that mathematicians often don't do this, but there is no good reason not to. Otherwise beginners may be confused about which things are vectors and which things are scalars. Gsspradlin (talk) 01:14, 2 May 2013 (UTC)gsspradlin

Orthonormal basis definition

I suggest changing "An orthonormal basis for an inner product space V is an orthonormal sequence whose algebraic span is V." to "An orthonormal basis for an inner product space V is an orthonormal sequence {e_k}_k which has V as the smallest closed subspace containing {e_k}_k. Equivalently, its closed linear span is V."--Matumba 11:10, 9 January 2007 (UTC)

I have put it another way. Charles Matthews 12:56, 9 January 2007 (UTC)

Cauchy-Schwarz Proof Error

The Cauchy-Schwarz proof may be important, but on the "mathematician's" conventions adopted on this page, it seems to me to be wrong. \lambda should be defined as <y,y>^{-1}<x,y> rather than <y,y>^{-1}<y,x> as at present. Then, for example, a term like <\lambda y, x> can be rearranged as \lambda <y,x> = <y,y>^{-1}<x,y>\overline{<x,y>} = <y,y>^{-1}|<x,y>|^2, as required later in the proof.

—The preceding unsigned comment was added by 88.111.140.233 (talk) 16:16, 18 March 2007 (UTC).

You're right. The article at one time used the physicist convention and during that time the proof was included.--CSTAR 18:21, 18 March 2007 (UTC)

Relating to definition of angle

Although it is hinted in the article that inner product (over Euclidean spaces) is related to intuitive definition of angle, and infact the inner product is used to define the angle (in the subsection "Norm"), it would be beneficial to show that this indeed corresponds in Euclidean spaces to the usual definition of angle given by arccos of base/hypotenuse. Ustad NY 13:11, 13 July 2007 (UTC)

Map symbolism

The symbols following the word "map" are unfamiliar to me. Can a link to an explanation of them be supplied? Unfree (talk) 20:22, 31 July 2009 (UTC)

Which symbols? The angle brackets? Or the mapsto arrow? Or the colon? The angle brackets are a conventional inner product notation. Everything else comes from the conventional symbols that define mathematical functions. —TedPavlic (talk/contrib/@) 12:19, 3 August 2009 (UTC)

HTML problems

This section does not render in Google Chrome: some symbols appear as empty boxes. I went into edit to see which HTML character it was; but it also appeared as a box. ~~ Dr Dec (Talk) ~~ 21:35, 20 December 2009 (UTC)

I have a similar problem. I guess the symbols ought to be brackets or something. User:KSmrq/Chars has an extensive list of HTML characters including ⟨ (&lang;). COuld you fix it this way? I don't have the time. Jakob.scholbach (talk) 21:56, 20 December 2009 (UTC)

Connection with metric

This section mentions several times that it's possible to induce a metric from an inner product, but neither this page nor the page on metrics shows this relation... —Preceding unsigned comment added by 131.155.68.160 (talk) 13:12, 21 February 2011 (UTC)

The first occurrence of the word metric is followed by a link on the word induced, and this link goes to an explanation - except that the section link is broken, and the linked article talks about the metric without saying that it's a metric, and in any case its talking about a metric induced by a norm, and no norm has been mentioned at the point of the cryptic link. So, yes, it's pretty bad, and a number of fixes are needed, both here and in the normed vector space article. --Zundark (talk) 16:32, 21 February 2011 (UTC)

Fixed, please see my edit. There is no need to write about metric here. A norm does the same trick better. 2andrewknyazev (talk) 00:54, 22 February 2011 (UTC)

Why does "Inner Product" redirect to this article?

This article is confusing without some prior knowledge of what the inner product is by itself. I can see how the concept of an inner product space is closely related to that of an inner product... but this article seems to be written with the assumption that the reader already knows about inner products. There is admittedly some mention of what an inner product does, but I have found no real explanation of what it is. It seems like there should be a separate article for "Inner Product"... or at least a section explaining it. — Preceding unsigned comment added by 79.246.47.200 (talk) 09:14, 1 May 2012 (UTC)

It seems to me that the article can hardly be improved in this respect. Defining an inner product cannot be done without first defining vectors and a vector space, and only then defining the the inner product, upon which the article focusses in its lead and first section (Definition). — Quondum^☏ 11:24, 1 May 2012 (UTC)

Thanks for the answer, though I still partially disagree. A basic understanding of vectors is certainly necessary. I'm not so sure about vector spaces. MathWorld does a pretty good job of giving the basic concept behind an Inner Product without first mixing it with that of a vector space. It was admittedly pretty easy to extract the definition of an inner product from this article once I had read the Wolfram equivalent, but only because I knew what to look for.

If the vast majority of visitors to this article are already familiar with vector spaces, then this is probably not a problem. (Since getting at the definition of the inner product becomes a relatively simple process of elimination in this case.) Maybe I am just a bit too far from my field. (Engineering) — Preceding unsigned comment added by 79.246.54.99 (talk) 07:51, 2 May 2012 (UTC)

This article should be accessible to mathematicians and engineers alike, so perhaps it needs a bit of a rewrite. Perhaps the use of the term "space" could be avoided in the early stages of the decription (even though familiarity with the properties of vectors implies familiarity with a vector space, even if not necessarily under that name); maybe merely referring to vectors would be adequate. Anyone want to tackle wording in this article to address this? — Quondum^☏ 13:02, 2 May 2012 (UTC)

"Naturally induces"

In the intro: "An inner product naturally induces an associated norm" What do the words "naturally" and "induces" mean in this context? Is it possible to "unnaturally induce" something? Does this sentence translate in plain english simply to: "An inner product implies an associated norm", or perhaps "An inner product provides a basis for calculating a norm" or something along those lines? Gwideman (talk) 21:18, 18 August 2012 (UTC)

It means that if you give me the inner product, I can use it to produce (in a natural way) a norm. Your two sentences together get close; the second sentence is problematic because of the use of "basis". This is explained more fully in the body of the article: "However, inner product spaces have a naturally defined norm based upon the inner product ...." Keeping in mind that this is just an introduction and the details are explained in the body, do you think there's a good way to clarify this jargon that doesn't make the sentence too much of a mess? --JBL (talk) 01:29, 23 August 2012 (UTC)

@JBL: Thanks for your comments. Yeah, got it regarding "basis"; obviously I was intending the everyday meaning. Anyhow, it's still a mystery to me what "naturally" means. Does it just trivially mean "oh look, combined with square root we have Pythagoras, which is the length of a vector in ordinary geometry, and hence it's a useful norm"? Or "naturally" mean something more profound? Gwideman (talk) 11:27, 1 September 2012 (UTC)

"Naturally" is a term that seems to be used extensively in mathematics, meaning (I infer as a non-mathematician) that in there is some sense essentially only one way to do it. Sometimes, that means that there may technically be any number of ways to do it, but that all those ways are equivalent/indistinguishable/isomorphic. So yes, it does mean something more profound, but I have yet to find a mathematically precise definition of the term. — Quondum^☏ 14:08, 1 September 2012 (UTC)

pre-Hilbert space confusion

From the intro: " An incomplete space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm, induced by the inner product, becomes a Hilbert space."

Huh? So there's a thing called a pre-Hilbert space, which is incomplete, and has an inner product, but because it has an inner product it also has a norm, and because it has a norm it is "complete with respect to the norm", and therefore it's a Hilbert space. So according to this sentence, there is no difference between a Hilbert space and a pre-Hilbert space, except perhaps the state of the explanation? Gwideman (talk) 21:26, 18 August 2012 (UTC)

You are correct until "and because it has a norm". Rather, because it has a norm it can be completed in this norm to give a different space called its completion, and the completion is a Hilbert space. --JBL (talk) 01:30, 23 August 2012 (UTC)

Thanks for the reply, and sorry about the delay on my part. The key is that the norm is used in the act ("completion") of creating a new space whose properties qualify it to be a Hilbert space. This is obfuscated by the current wording "becomes a Hilbert space". There is actually no space that formerly was not a Hilbert space, and then presto "becomes" a Hilbert space. Thanks for clearing that up. Gwideman (talk) 00:37, 28 August 2012 (UTC)

Shouldn't it be "any field"?

The definition in this article starts with: "In this article, the field of scalars denoted F is either the field of real numbers or the field of complex numbers."

But isn't an inner product space defined for any field?

For instance http://mathworld.wolfram.com/InnerProduct.html says: "This definition also applies to an abstract vector space over any field."

Klaas van Aarsen (talk) 11:28, 18 November 2012 (UTC)

The definition given in MathWorld isn't even correct for a vector space over the complex numbers (since axiom 3 is wrong), and for an abstract base field it's meaningless (since the expression used in axiom 4 is undefined). --Zundark (talk) 13:03, 18 November 2012 (UTC)

I dont know if the article has been changed since you made this comment, but it currently reads that F denotes the field of reals or complex, specifically in this article... not that the field always has to be reals or complex. 50.35.97.238 (talk) 20:21, 6 November 2018 (UTC)

Difficult to Read

I'm not really a mathematician, but this is very difficult to understand! I feel like this ought to be a simple operation, but it is very difficult to tell what the inner product is actually doing based on how the article is written. I'm sure everything is correct, but having never seen this before I have no idea what it is talking about. Could someone make the language a little clearer and easier to understand? Spirit469 (talk) 18:08, 26 February 2013 (UTC)

Perhaps you would like to look at euclidean space where the inner product really is a simple operation? And for not so simple inner products, consult Sobolev spaces where the inner product involves the weak derivatives of functions? In an abstract Hilbert space, the inner product really just is a blackbox where you put two vectors (points in a vector space, not necessarily column vectors) in and get a real or complex number out. With certain restrictions on linearity etc. that allows, e.g., to define the angle of two vectors.--LutzL (talk) 18:13, 26 February 2013 (UTC)

I'm having trouble with it, too. If it's possible to give an example of an inner product at its simplest, using numbers, I suspect it would help a lot. Bxb Grxmmxn (talk) 19:05, 2 March 2013 (UTC)

I think the article you are looking for is dot product, which is the commonest and simplest instance of an inner product. That article is linked in the first paragraph, and I would expect anyone looking for it will find that article first – "dot product" is the name most often used when it's being studied at high school level. That does include numerical examples.--JohnBlackburne^words_deeds 19:43, 2 March 2013 (UTC)

Remarks about using fields other than C or R are inadequate and contradictory

In the "Remark", under the "Definition", it is stated that it is necessary to restrict the basefield to R or C. This statement is contradicted in the same paragraph by the statement that any quadratically closed subfield of R or C will suffice. It is stated that "The basefield has to have additional structure, such as a distinguished automorphism." No reason or reference is given for this assertion (either for the fact that additional structure is needed, or exactly what that additional structure has to be). "Distinguished automorphism" is not defined, and there is no link to a definition. Gsspradlin (talk) 16:03, 2 May 2013 (UTC)

Incompatible example

According to the definition a property of an inner product is "Linearity in the first argument". In the example section there is an example of an inner product:

$\langle \mathbf {x} ,\mathbf {y} \rangle :=\mathbf {x} ^{*}\mathbf {M} \mathbf {y} .$

I think that this example is incompatible with the definition. I believe that in order to be linear in the first argument, it should read

$\langle \mathbf {x} ,\mathbf {y} \rangle :=\mathbf {y} ^{*}\mathbf {M} \mathbf {x} .$ — Preceding unsigned comment added by Panagiotis.niavis (talk • contribs) 17:49, 12 May 2013 (UTC)

I'm affraid you are right! Bdmy (talk) 20:25, 12 May 2013 (UTC)

Confusion between conventions in math and in physics

When the current version of the article states the following:

• Linearity in the first argument:

$\langle ax,y\rangle =a\langle x,y\rangle .$

the reader should probably be cautioned that this is different from the convention common to physics textbooks:

$\langle ax,y\rangle ={\overline {a}}\langle x,y\rangle .$

The article uses the pure math textbook convention expressed in notation commonly used only among physicists, this may confuse some readers.

Cool dude ragnar (talk) 09:37, 30 October 2013 (UTC)

Did You read the remarks at the end of the definition, starting with "Some authors, especially in physics and matrix algebra, prefer..."? --LutzL (talk) 10:04, 30 October 2013 (UTC)

Of course, if one wants to weight the points of view, the mathematical one is slightly more "wrong". Mathematicians like to write the basis decomposition as $v=\sum \nolimits _{k=1}^{n}\langle v,e_{i}\rangle \cdot e_{i}$, expressing the preference to write the coefficient before the vector and to have the two occurrences of the basis vector close together. For this to work one needs complex linearity in the first argument. However, from a matrix point of view, this can be most consistently rewritten as $v=(e_{1},\dots ,e_{n})\cdot (x_{1},\dots ,x_{n})^{\top }$, where clearly the coefficient comes after the basis vector, and to have the basis vectors typographically close together the scalar product should be $x_{k}=\langle e_{k},v\rangle =e_{k}^{*}v$, which would need complex linearity in the second argument.--LutzL (talk) 10:18, 30 October 2013 (UTC)

Dear LutzL (talk), thanks for your informative reply.

The remark is at the end of a list of lemmas that follow from the 3 axioms. I - and I believe many readers will do the same - assumed the actual definition ended with the line:

$\langle x,x\rangle \geq 0$ with equality only for $x=0.$

May I suggest moving the caveat in among the list of axioms or immediately after, and splitting off the discussion of lemmas into a subsection captioned "Basic lemmas"?

Cool dude ragnar (talk) 15:37, 30 October 2013 (UTC)

History section

Could we have a paragraph or so regarding the history of the inner product? When and by whom it was defined, and in what context? Thanks.CountMacula (talk) 04:57, 25 November 2013 (UTC)

Positive definiteness and the order relation on the scalars

As the inner product maps to the underlying field of scalars, and the complex field lacks an order relation, how does the inequality in the axiom on positive definiteness make sense? Should the left-hand side of the inequality be the _norm_ of the inner product of the vector with itself, instead of simply the inner product?CountMacula (talk) 05:31, 25 November 2013 (UTC)

Note that the conjugate symmetry implies that $\langle x,x\rangle$ is real. --Zundark (talk) 13:27, 25 November 2013 (UTC)

Ha ha, okay, thank you.CountMacula (talk) 21:35, 25 November 2013 (UTC)

Why must the inner product map to reals or complex numbers at all? Why cannot F be a general field? Do we have a generalized definition for inner products or does the field have to be specific to reals? 50.125.86.70 (talk) 19:18, 21 April 2019 (UTC)

Strengthening to separability ⇔ countable basis?

From article:

Theorem. Any separable inner product space V has an orthonormal basis.

Can this be strengthened? For Hilbert spaces the following is true:

Theorem. A Hilbert space H is separable if and only if H has a countable orthonormal basis. YohanN7 (talk) 13:40, 3 April 2014 (UTC)

Wait what? In any Hilbert space, the basis has to be at least continuum in cardinality. 62.77.218.240 (talk) 17:13, 27 October 2020 (UTC)

POV article

As suggested by a few threads above, the restriction to positive-definiteness seems to be discipline-specific. Just because some fields (e.g. quantum mechanics) choose to restrict the types of inner products that they work with does not mean that their restrictions should be imposed on the whole subject of linear algebra. Irving Kaplansky (1969). Linear Algebra and Geometry. §Inner product spaces defines inner product spaces clearly in the general sense:

• An arbitrary base field is permitted. Finite fields, including of characteristic 2, are covered.
• An arbitrary symmetric bilinear form is permitted.
• There is no restriction to positive-definiteness, not even to nondegeneracy. A degenerate example is given.

Kaplansky goes on to describe a broadening of the theory to alternating and Hermitian forms, but it is not clear to me whether he is including these under the definition of inner product spaces, though he does indicate that it can make sense to include such generalizations from the start. So: is anyone going to claim that Kaplansky is not a notable secondary source? By implication, unless the contrary is demonstrated, the article must be updated to the more general definition. —Quondum 16:56, 27 July 2014 (UTC)

Why not just list Kaplansky in the Alternative definitions, notations and remarks section? What he calls an inner product, most everybody else calls a symmetric bilinear form. YohanN7 (talk) 18:24, 27 July 2014 (UTC)

This would depend on the majority view of the discipline in which it is presented. If the article started "In physics, ...", I'd agree with you. Either change that (though I think you'd find that a hard sell: mathematicians might say it belongs in a mathematical discipline), or bring in notable references from general linear algebraists. The references are heavily skewed to Hilbert spaces and quantum mechanics; in fact, this article few references. Some mathematicians also tend to put normed spaces in primary position, so I understand that there will still be tension between the ideas even regarded within the scope of mathematics. The question becomes: within mathematics, how is the term used generally? I've provided a counterexample that suggests a presentation in which the concept of an inner product that naturally becomes positive-definite when applied in the context of a Hilbert spaces. In a way, this might be the usual tension between: do we present the familiar scope first, then give a "generalizations" section, or do we define the concept as the more general case, and then expand on the more familiar case? I think it comes down to what mathematicians generally consider the term to mean. —Quondum 18:49, 27 July 2014 (UTC)

GTM 135 uses exactly the definition given in the article. Edit: As does GTM 96. I don't think "most general" has automatic precedence over "most common". YohanN7 (talk) 19:04, 27 July 2014 (UTC)

Agreed on precedence. In this context, I would like to see this as "most common amongst mathematicians", or at least generally. At the moment, references seem to be skewed to physics. —Quondum 21:44, 27 July 2014 (UTC)

Semi-inner product

An edit summary from a recent edit:

Dropped the reference to the semi-inner product. Eliminating the 2nd requirement in the Positive-definite definition does not agree with the wiki definition of semi-inner product.

Fair enough. But what is then the thing called? John B. Conway calls it a semi-inner product in his A course in fuctional analysis. YohanN7 (talk) 11:53, 8 January 2016 (UTC)

Large-scale edits reverted

Mgkrupa did a lot of edit on this page. Yesterday, I restored the version of § Definition that predated his edits, because it was correct and clearer, while Mgkrupa version was confusing by hidding the main facts behind comments that are either out of scope, or original research, or both.

Mgkrupa has essentially restored his version and moved his editorial comments to a new section § Remark, that is inacceptable in Wikipedia because of WP:EDITORIAL and WP:SYNTHESIS. He also restored or added inacceptable edits such as the introduction of grammatical errors (removing the article before "inner product space"), unusual notation ($\mathbb {F}$ for a partially unspecified field), ordering the axioms of inner product spaces by numbering them, etc.

So, I have reverted all this stuff. Per WP:BRD, editing the article again must be avoided without explaining the reasons of the edit (never done by Mgkrupa), and without getting a consensus on the talk page. D.Lazard (talk) 09:22, 22 December 2021 (UTC)

"introduction of grammatical errors" Oops. A minor problem with an easy fix. Everyone makes a minor mistake here and there. (e.g. you misspelled "hidding")

"ordering the axioms of inner product spaces by numbering them" Is there a policy against this? I am not aware of any. If so them change <ol> to <ul>. Problem solved.

"$\mathbb {F}$ for a partially unspecified field" Is there a policy against this notation? I am not aware of any. Is this is a problem then this can be fixed by find+replace all.

Some of the proofs I took from the references that I added. I add those proofs for readers who might be interested in reading them.

"editing the article again must be avoided without explaining the reasons of the edit (never done by Mgkrupa), and without getting a consensus on the talk page." Same applies to you.Mgkrupa 09:33, 22 December 2021 (UTC)

@D.Lazard:You have not replied to any of my responses. List what objections you have against this version of the article. If you only list trivial objections OR if you do not response in a reasonable amount of time then I will revert your revert.Mgkrupa 13:52, 22 December 2021 (UTC)

Per WP:BRD, it is to you to show that your edits improve the article. In particular, it is to you to provide sources that show that the discussions referred to as "remarks" or "motivations" are not WP:Original synthesis. Also, MOS:VAR states that if a style is acceptable, it is forbidden to change it without explicit strong reason. This applies to the use of $\mathbb {F} ,$ to the change of list format from bulleted to numbered, to the use of html tags (<ol>) for lists, and to many changes that are individually minor, but all together make the article harder to read for non-specialists.

"motivations" is gone. The information in the "remarks" section was not added by me; it was there before I edited this article. (added later: to be clear, I'm referring to my latest version here 09:38, 22 December 2021; not to your versionMgkrupa 15:35, 22 December 2021 (UTC)).

So are we in agreement to replace "$\mathbb {F} ,$" with "$F$"? Are we in agreement to replace "<ol>" with "<ul>"?

Please list your other objections/suggestions. And please refrain from any more edit warring.Mgkrupa 14:45, 22 December 2021 (UTC)

I agree to not edit this article, except to revert edits that are not a clear improvement or are not be validated by a consensus in this page. If you revert my revert, this will be WP:edit warring, and this may lead you to be blocked for editing. D.Lazard (talk) 14:37, 22 December 2021 (UTC)

I would also like to point out that @D.Lazard: violated the the three-revert rule with these three reverts

1. 21:18, 21 December 2021
2. 08:50, 22 December 2021
3. 10:28, 22 December 2021

Mgkrupa 14:37, 22 December 2021 (UTC)

It is very difficult to provide a third opinion on this dispute because there are literally hundreds of edits involved. I will try to provide an opinion on the differences I see between the last two versions of the article ([1] and [2], henceforth 1 and 2; 1 is the current version and D. Lazard's version). Here are some differences, including those discussed above:

• Cosmetic

• I think R and C should be mathBB, F should not be.
• 1's version of the definition section is better: unordered lists, no bold labels on the right
• 2 formats more of the math as html, e.g {{math...}}. I think the MOS policy is not to make sweeping changes to the math formatting without a good reason? So is D. Lazard's version the "original" here? Mgkrupa, do you want to argue for this change? Should we just agree to a policy for this article and make it uniform?

• Substantive

• I see a long Remarks section in 2. I think this content is interesting but it should not be at that place in that form. For the question about which position should be linear, the sentence or two in Note 1 seems sufficient, and if there should be more content about this, it should be further down, at least under the examples and basic properties. The part about the base field can be discussed (ideally more concisely) in the section about "Real and complex parts of inner products", or perhaps immediately before that subsection. But it shouldn't delay the examples and basic properties.
• 2 has a sentence about dot products at the start of "some examples". This seems like it should be moved into the Euclidean space subsection, where there should be an example of using the dot notation.
• 2 has a properties section at the top of "Basic results, terminology, and definitions" about L-semi-inner-products and Hamel basis, which are not topics I'm familiar with. This definitely shouldn't be the first subsection of this section, but I don't have an objection to it being moved to the bottom. (D. Lazard, do you want to specifically argue against this content being anywhere on the page?)
• 2 has a section "Completeness and Hilbert Spaces". This seems fine to me. D. Lazard, do you have something against it?

My suggestion is to use 1 as a baseline and try to reach a compromise from there. Mgkrupa, are there other edits or specific disputes that I'm missing that way? Is there another version in the history that I should be looking at? Danstronger (talk) 14:16, 23 December 2021 (UTC)

For the record my recent edits consisted only to restore the version of § Definition, dated from September 2020, 5, just before the first edit of Mgkrupa (see the diff. [3]), and revert Mgkrupa's reverts. Otherwise, I have not edited the article recently. This does not mean that I find the section and the remainder of the article perfect. Only that the old version is much better than Mgkrupa's one.

Just now, I remark that after his first revert, Mgkrupa did several edits that I reverted with my subsequent reverts. This includes the new section on completeness and Hilbert spaces; see below. Also, Mgkrupa has edited other parts of the article. I have not checked them, and thus I have no opinion on which edits are improvements, and which are not. This is because I do not want to spent too much time on this article, and if the lead and the definition section are correct, the remainder of the article is less important for most users.

This being said,

• I agree that R and C should be replaced with $\mathbb {R}$ and $\mathbb {C} .$
• There is a section § Hilbert space inside § Examples.
• About Hilbert spaces: There is a longstanding subsection § Hilbert space in § Some examples, and Mgkrupa added recently a section "Completeness and Hilbert spaces". IMO, both are badly written, and they deserve to be merged. However, this may hardly be done without cleaning up the awful section § Basic results, terminology, and definitions.

D.Lazard (talk) 10:03, 24 December 2021 (UTC)

@D.Lazard: I see. I have changed the Rs and Cs. I'm confused about the Remarks section. You complained about it but Mgkrupa said he did not add it (he just reorganized it into a higher-level section), and your version (10:28, 22 December) has essentially the same content in an "Alternative definitions, notations and remarks" section (plus the sentence about dot products). Did you want this content removed? Danstronger (talk) 16:02, 26 December 2021 (UTC)

My opinion is that, before Mgkrupa's edits the article had many serious issues, that are not fixed by Mgkrupa's edits. Calling a section "Remark" goes against WP:NOTTEXTBOOK and WP:SYNTHESIS, since such a title suggest that the content is an editorial synthesis. Indeed the second subsection of "Remark" (that is, "Base field") is pure original synthesis and does not belong to the article. On the other hand, the other subsection, called "Notation and the antilinear argument" deserve to appear in its own section, called "convention variant", with an improved writing.

Here is an important issue: A non-expert reader must easily know which properties are specific to inner product spaces, and which are general properties of normed spaces. He must also clearly understand that giving an inner product is equivalent to give a definite Hermitian form. Presently, all these properties are mixed in a confusing and unstructured way. This was already the case with the older version, but is worse after Mgkrupa edits. D.Lazard (talk) 21:35, 26 December 2021 (UTC)

Ok. I agree the part about base field is OR, and the dot product is mentioned in the first paragraph of the lead already. I deleted them, and most of the stuff about bra-ket notation, leaving just a couple of sentences in a "convention variant" section, which mostly still needs to be written, but I think it's better as it is than with all the confusing or unnecessary content that was there before. (If you think I overdid it, I'm happy to add back the stuff about bra-ket notation; it just seemed too confusing to be useful as it was.) Danstronger (talk) 23:30, 26 December 2021 (UTC)

Hi everyone. The page User:Mgkrupa/sandbox contains a rough idea of the changes I'd like to see. I have not polished it yet (due to time) and any small mistakes would (of course) be fixed before adding it to the actual article. Parts of it also need to be reworded. One more thing that I'd like to include is an answer to the common question "What are complex inner products conjugate symmetric?" I'll do these changes and also respond to some comments/questions above once I have more time. Mgkrupa 01:05, 28 December 2021 (UTC)

My proposed changes are here: User:Mgkrupa/sandbox. Any suggestions or objections Danstronger or D.Lazard? Mgkrupa 23:37, 29 December 2021 (UTC)

I'm fine with merging the sentence "In this article, the field of scalars denoted F..." into the following sentence and making F mathBF. I generally don't like the other changes to "Definition" -- making it more complicated, adding the small/bold/right-justified/parenthetical labels. You took out some stuff about additivity; I'm fine with this removal. But your new section on "alternative definitions" seems unnecessary and verbose. I could see having a sentence or two here about what a "hermitian form" is or whatever, but not three paragraphs. Actually the original version has the couple of sentences I'm envisioning, in the middle of "Elementary Properties".

It doesn't make any sense for the first mention of "Hilbert" on this page to be a non-Hilbert example without explaining what a Hilbert space is. I think the 1-d examples should be cut and the examples should be: R^n, C^n, an actual Hilbert example (starting with one sentence explaining that a Hilbert space can be infinite-dimensional but it must be complete), and then maybe the non-Hilbert example. I think then the "completeness and hilbert spaces" section will be unnecessary. Danstronger (talk) 00:29, 30 December 2021 (UTC)

I made some changes to User:Mgkrupa/sandbox based on your (Danstronger) suggestions. I'm fine with cutting the 1-d examples and your suggestions about how to introduce Hilbert spaces. But there should be a discussion somewhere about how an inner product space relates to its completion. Mgkrupa 02:20, 30 December 2021 (UTC)

I still think most of the changes you're proprosing to the definition section make it worse. The overuse of bold and italics is not good (see WP:MOSBOLD, for one). The elaboration of positive definite into definite and positive semi-definite is unnecessary. The discussion of Hermitian forms is still too long. Maybe a sentence could be added about the alternative definition of an inner product space, but not two paragraphs.

The definition of a Hilbert space shouldn't come before the Euclidean space examples. Defining a Hilbert space as a Banach space is unhelpful. Danstronger (talk) 05:39, 30 December 2021 (UTC)

Removed redundant sentence

Anyone opposed to removing the following sentence?

In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C.

Mgkrupa 09:40, 22 December 2021 (UTC)

@Mgkrupa: That sentence is necessary to introduce the "variable" F. Danstronger (talk) 16:03, 26 December 2021 (UTC)